Find The Value Of $i^{703}$.A. 1B. IC. $-1$D. $-i$

Introduction

The imaginary unit, denoted by $i$, is a fundamental concept in mathematics, particularly in algebra and geometry. It is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. The powers of $i$ follow a cyclical pattern, which is essential to understanding the behavior of complex numbers. In this article, we will delve into the world of complex numbers and explore the value of $i^{703}$.

The Cyclical Pattern of Powers of $i$

The powers of $i$ follow a cyclical pattern, which can be expressed as:

$$i^1 = i$$

$$i^2 = -1$$

$$i^3 = -i$$

$$i^4 = 1$$

This pattern repeats every four powers, which is known as the period of the powers of $i$. Understanding this cyclical pattern is crucial in determining the value of $i^{703}$.

Determining the Value of $i^{703}$

To find the value of $i^{703}$, we need to determine the remainder when $703$ is divided by $4$. This is because the powers of $i$ repeat every four powers, and the remainder will indicate which power of $i$ we are dealing with.

Calculating the Remainder

To calculate the remainder, we can use the modulo operator, which is denoted by $\%$. In this case, we need to find the remainder when $703$ is divided by $4$.

$$703 \mod 4 = 3$$

This means that the remainder is $3$, which indicates that we are dealing with the third power of $i$.

Finding the Value of $i^{703}$

Now that we know the remainder is $3$, we can determine the value of $i^{703}$. Since the remainder is $3$, we are dealing with the third power of $i$, which is:

$$i^3 = -i$$

Therefore, the value of $i^{703}$ is $-i$.

Conclusion

In conclusion, the value of $i^{703}$ can be determined by understanding the cyclical pattern of powers of $i$ and calculating the remainder when $703$ is divided by $4$. The remainder indicates which power of $i$ we are dealing with, and in this case, it is the third power of $i$. Therefore, the value of $i^{703}$ is $-i$.

Answer

The correct answer is:

D. $-i$

Discussion

This problem is a great example of how understanding the cyclical pattern of powers of $i$ can help us determine the value of complex numbers. It also highlights the importance of calculating the remainder when dividing by $4$ to determine which power of $i$ we are dealing with.

Additional Examples

Here are a few additional examples to illustrate the concept:

• $i^{701} = i^1 = i$
• $i^{702} = i^2 = -1$
• $i^{703} = i^3 = -i$
• $i^{704} = i^4 = 1$

These examples demonstrate how the powers of $i$ repeat every four powers, and how the remainder when dividing by $4$ can help us determine the value of complex numbers.

Conclusion

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Frequently Asked Questions

Q: What is the cyclical pattern of powers of $i$?

A: The powers of $i$ follow a cyclical pattern, which can be expressed as:

$$i^1 = i$$

$$i^2 = -1$$

$$i^3 = -i$$

$$i^4 = 1$$

This pattern repeats every four powers, which is known as the period of the powers of $i$.

Q: How do I determine the value of $i^{703}$?

A: To find the value of $i^{703}$, you need to determine the remainder when $703$ is divided by $4$. This is because the powers of $i$ repeat every four powers, and the remainder will indicate which power of $i$ we are dealing with.

Q: What is the remainder when $703$ is divided by $4$?

A: The remainder when $703$ is divided by $4$ is $3$. This means that we are dealing with the third power of $i$.

Q: What is the value of $i^{703}$?

A: Since the remainder is $3$, we are dealing with the third power of $i$, which is:

$$i^3 = -i$$

Therefore, the value of $i^{703}$ is $-i$.

Q: Why is it important to understand the cyclical pattern of powers of $i$?

A: Understanding the cyclical pattern of powers of $i$ is crucial in determining the value of complex numbers. It helps us to identify which power of $i$ we are dealing with, and therefore, determine the correct value.

Q: Can you provide more examples of how to find the value of $i^n$?

A: Yes, here are a few additional examples:

• $i^{701} = i^1 = i$
• $i^{702} = i^2 = -1$
• $i^{703} = i^3 = -i$
• $i^{704} = i^4 = 1$

These examples demonstrate how the powers of $i$ repeat every four powers, and how the remainder when dividing by $4$ can help us determine the value of complex numbers.

Q: What are some real-world applications of complex numbers?

A: Complex numbers have numerous real-world applications, including:

• Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
• Control systems: Complex numbers are used to analyze and design control systems.
• Quantum mechanics: Complex numbers are used to represent wave functions and analyze the behavior of particles.

Conclusion

In conclusion, the value of $i^{703}$ can be determined by understanding the cyclical pattern of powers of $i$ and calculating the remainder when $703$ is divided by $4$. The remainder indicates which power of $i$ we are dealing with, and in this case, it is the third power of $i$. Therefore, the value of $i^{703}$ is $-i$.